Is the value of $\sin(\frac{\pi}{n})$ expressible by radicals?

As was alluded to in comments this relates to the nth roots of unity. First of all you distinguish between 'primitive' and non-primitive. Then you look if those primitive roots are expressible in radicals and so on. The case where they are not happens for example with the Casus irreducibilis, which sometimes (!) happens in trisecting the angle (but never in bisecting the angle). Long story short it turns out for example that for $0<=k<=2^n$ all $\sin(2\pi k/2^n)$ and $\cos(2\pi k/2^n)$ are expressible in radicals!

You'll find an excellent start here: http://www.efnet-math.org/Meta/sine1.htm